I have the following
(a) Define a sketch $s_{\mathbf{Grp}}$ and a equivalence functor $$E: \mathbf{Grp}\to \mathbf{Mod}(s_{\mathbf{Grp}},\mathbf{Set})$$ (b) Knowing that finite limits commute with filtered colimits in $\mathbf{Set}$, use the result in (a) to prove that they also commute in $\mathbf{Grp}$.
(c) Prove that $\mathbf{Ab} \backsimeq \mathbf{Mod}(s_{\mathbf{Grp}},\mathbf{Grp})$
This is my first exercise working with sketches. I've been told that even simple examples with sketches gives us a big amount of work. In fact, I couldn't do (a) at a first glance. But I found a useful example at nlab's sketch article. Example 3.2 especifies the directed graph, diagrams, cones and cocones of a sketch which has unital magmas as models (sets with a binary operation which has a two sided unit).
So I thought taking this same sketch and "interpreting" the arrows $e$ as the identity of a group and $m$ as its multiplication, all this via the equivalence functor the exercise request us to build. But I don't even know how to finish the construction of $E$ and in fact I don't see why it will be an equivalence functor at all.
Could you help me? Also is there any result that I'm missing on (b)? Because I think this shouldn't be so difficult.
The case of semigroups is spelled out in all details in Sec 2 of Chap 7 of Barr and Wells's Category theory for Computer Science.